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Questions tagged [central-extensions]

Use this tag for questions about short exact sequences of groups 1 → A → E → G → 1 such that A is in the center of E.

A central extension of a group G is a short exact sequence of groups 1 $\rightarrow$ A $\rightarrow$ E $\rightarrow$ G $\rightarrow$ 1 such that A is in the center of E. The set of isomorphism classes of central extensions of G by A (where G acts trivially on A) is in one-to-one correspondence with the cohomology group H$^2$(G, A).

Examples are found in the theory of projective representations in cases where the projective representation cannot be lifted to an ordinary linear representation.

In the case of finite perfect groups, there is a universal perfect central extension.

Similarly, the central extension of a Lie algebra $\mathfrak g$ is an exact sequence 0 $\rightarrow \mathfrak a \rightarrow \mathfrak e \rightarrow \mathfrak g \rightarrow$ 0 such that $\mathfrak a$ is in the center of $\mathfrak e.$

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Central extensions versus semidirect products

Consider an extension $E$ of a group $G$ by an abelian group $A$. $$1 \to A \overset{\iota}{\to} E \overset{\pi}{\to} G \to 1$$ Two special kinds of extensions are: Central Extensions: $A$ is contained in the centre of $E$. Semidirect products:…
Mike F
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What exactly is a universal central extension?

Let $G$ be a group. In An introduction to homological algebra, Chapter 6.9 Weibel defines a universal central extension as a central extension $$0 \to A \to X \to G \to 1,$$ which is initial with respect to all central extensions. In other words, if…
red_trumpet
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Is every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ a square element in finite spin group ${\rm Spin}_n^{\epsilon}(q)$?

A. I wonder if every generator of $Z({\rm Spin}_n^{\epsilon}(q))$ is a square element in ${\rm Spin}_n^{\epsilon}(q)$? B. When $Z(\Omega_{2m}^{\epsilon}(q))\cong C_2$, is the generator of $Z(\Omega_{2m}^{\epsilon}(q))$ a square element in…
Yi Wang
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Structure of affine Lie algebras

It is well-known that to every simple Lie algebra $\mathfrak{g}$ one can associate an affine Kac-Moody algebra by a double extension (once by a 2-cocycle and once by a derivation). One can then show that this algebra is the Kac-Moody algebra…
NDewolf
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Why does the definition of cocyle in a central extension of a Lie algebra work?

I am currently reading Edward Frenkel's "Langlands Correspondence for Loop Groups", freely available here: https://math.berkeley.edu/~frenkel/loop.pdf. In the appendix $A.4$ he describes central extensions of Lie algebras: let $\mathfrak{g}$ and…
toyr99
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Group Cohomology, Module Extensions, and Group Extensions, and $Ext^2_{\mathbb{Z}G}(\mathbb{Z},A)$

I've read that for some $G$-module $A$, group cohomology can be defined as $$H^{n}(G,A)=Ext^{n}_{\mathbb{Z} G}(\mathbb{Z},A).$$ I've also read that for two $R$-modules $C,D,$ $Ext^{n}_{R}(C,D)$ can be viewed as the equivalence classes of $n$-fold…
Connor Mooney
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What is the explicit central extension given by the prequantization procedure (from functions on phase space to vector fields)?

The Question: The Lie algebra of functions on phase space (under the Poisson bracket) is a central extension of the Lie algebra of Hamiltonian vector fields on phase space (under the vector field Poisson bracket). This is because constant functions…
user1379857
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Virasoro algebra question: Is there a two-surface in Diff($S^1$) with a non-zero integral over the cocycle in $H^2(\mathfrak{g}, \mathbb{C})$?

I am a physicist so forgive me if this question doesn't make sense. You can start off by defining the Witt algebra, which I'll call $\mathfrak{g}$, as the complexified Lie algebra of vector fields on $S^1$. $$ \mathrm{Vect}(S^1) \otimes…
user1379857
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Reference request: Universal central extension of $\operatorname{PSl}_2(\mathbb Z)$ is the braid group $\mathcal B_3$.

$\DeclareMathOperator{\PSl}{PSl}$ According to Wikipedia, the universal central extension of $\PSl_2(\mathbb Z)$ is given by the braid group $\mathcal B_3$ on three strands, \begin{equation} \label{extension} \tag{1} 0 \to c \mathbb Z \to \mathcal…
red_trumpet
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Does $\mathbb R \overset{\iota}{\hookrightarrow} \mathfrak{aff}(1,\mathbb R) \overset{\pi}{\twoheadrightarrow} \mathbb R$ Lie algebra extension exist?

I think I have an example for a $\mathbb R \overset{\iota}{\hookrightarrow} \mathfrak{aff}(1,\mathbb R) \overset{\pi}{\twoheadrightarrow} \mathbb R$ Lie algebra extension. $$\iota:\mathbb R\to \mathfrak{aff}(1,\mathbb R): x\mapsto \begin{pmatrix}0 &…
mma
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Group Cohomology and Pontryagin Duality

My question is related to this question, which I tried to post an answer. I think it is better to ask a question directly. My question is from the book "Foundations of Quantum Theory: from classical concepts to operator algebras" by Klaas Landsman,…
Xenomorph
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Connection between central extensions and universal covering groups for projective representations

Disclaimer, I'm a physics student, so some of my arguments might be a little bit sloppy. I'm trying to find a somewhat straightforward general argument to explain why we need central extensions when working with quantum symmetry groups. Here's what…
justsomethingidc
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Nilpotent groups can be constructed by means of abelian groups.

I am studying A course in theory of groups by Robinson. When defining the central extension of groups, the author says that Every nilpotent group can be constructed from abelian groups by means of a sequence of central extensions. What do I…
MANI
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What controls the central extensions of Lie algebras when the Lie Group is not compact?

If the Lie algebra $\mathfrak{g}$ can be realized as the tangent space of a compact Lie group $G$, then all the possible central extensions of $\mathfrak{g}$ are in one to one correspondence which the second de Rham cohomology group of $G$. The…
user1379857
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Basic assertion about central extensions

I'm having trouble verifying what ought to be a relatively simple detail in a proof of Milnor's book on algebraic K-theory, in the section on universal central extensions. Here is the set up. Let $G$ be a perfect group ($G = [G,G]$), and let…
Joshua Ruiter
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